A Differentially Private Quantum Neural Network for Probabilistic Optimal Power Flow

We consider a radial distribution grid with controllable distributed energy resources (DERs). A distribution system operator (DSO) is responsible for managing the DERs and controlling the power supply from the main grid, ensuring that grid security constraints are satisfied. The grid is represented as a graph $\mathcal{G}=(\mathcal{B},\mathcal{L})$, where $\mathcal{B}$ is the set of buses and $\mathcal{L}$ is the set of lines. Each bus in $\mathcal{B}$ is indexed by $i\in\{1,\dots,N\}$ and each line in $\mathcal{L}$ is denoted as $l=(j,k)$. Suppose each controllable DER sited at bus $i$ is with cost coefficient $c_{i}$, then the deterministic OPF problem is formulated as:

where the objective is to minimize the total operation cost subject to branch flow equations (2)-(5) adopted in [10]. $P^{G}_{i}$ denotes the generation power of controllable DER sited in bus $i$. The power $p_{i}$/$q_{i}$ represents the active/reactive power at bus $i$, $P_{jk}$/$Q_{jk}$ is the active/reactive power flow on line $l=(j,k)\in\mathcal{L}$, $r_{jk}$/$x_{jk}$ denotes the resistance/reactance of line $l=(j,k)$, $\ell_{jk}$ is the squared current magnitude of line $l=(j,k)$, $v_{j}$ is the squared voltage magnitude at bus $j\in\mathcal{B}$, and $\underline{\bullet}$ and $\overline{\bullet}$ represent the lower and upper bounds of the respective variables.

Suppose there are wind and solar generation in the distribution grid. To account for the uncertainties inherent in wind and solar generation, as well as load demand variability in the distribution grid, we extend the deterministic OPF into a probabilistic OPF formulation as follows:

where $\boldsymbol{\xi}$ represents the set of random variables capturing system uncertainties (i.e., solar generation, wind generation, and load demand), while $\boldsymbol{u}$ denotes the same set of decision variables as in the deterministic OPF. The objective function $\mathbb{E}_{\boldsymbol{\xi}}[\min_{\mathbf{u}}f(\mathbf{u},\boldsymbol{\xi})]$ represents the expected minimum operational cost, where the operational cost is minimized for each uncertain scenario, and the expectation is taken over the distribution of these optimal operational costs. Constraint (10) represents the branch flow equations, and constraint (11) enforces the security limits on $\boldsymbol{u}$.

A classical method to solve the above probabilistic OPF problem is by the Monte Carlo simulation [11]. It draws samples of uncertain variables from the distributions, solves the deterministic OPF, and then aggregates the results. Despite the accuracy, Monte Carlo methods are computationally intensive. To accelerate the calculation of each sample, we leverage the computational power from quantum computing.

Quantum computing processes the information with qubit, i.e., the quantum bit. Unlike classical bits, which can only exist in a distinct state of either 0 or 1, qubits can exist in both states simultaneously due to the property of quantum superposition. Mathematically, the state of a single qubit can be represented as:

where $|0\rangle=\begin{pmatrix}1\\ 0\end{pmatrix}$ and $|1\rangle=\begin{pmatrix}0\\ 1\end{pmatrix}$ are the basis states in Hilbert space and $\alpha$, $\beta\in\mathbb{C}$ are complex amplitudes satisfying the normalization condition $|\alpha|^{2}+|\beta|^{2}=1$.

When solving practical problems, we use multiple qubits together by creating correlations between them. Such correlations are called entanglement. When qubits are entangled with each other, operations on any one of the qubits perform corresponding parallel operations on the other qubits. Therefore, this entanglement contributes to the potential exponential speedup that quantum computers may possess over classical computers for certain algorithms.

The quantum operations are performed by quantum gates. Quantum gates are unitary matrices in Hilbert space. For example, an elementary quantum gate, Pauli-X gate is denoted by $\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$, which flips the state of a qubit. When applying quantum gates $U$ on the qubits, the quantum states unitarily change from initial state $|0\rangle$ to a new state $|\psi\rangle=U|0\rangle$, which is the unitary evolution of qubits. By designing appropriate quantum gates, qubits evolve toward the expected state, which can then be measured to collect the final result.

To quantify and control the privacy risks of the customer loads, we adopt the framework of differential privacy [13]. It protects privacy by ensuring the outputs from two similar inputs are in-distinguishable. Before introducing the mechanism of differential privacy, we briefly introduce the meaning of “similarity”.

Suppose a load dataset is represented by $D\in\mathbb{R}^{n}$ with $i$-th element $d_{i}$ describing the active load of node $i$. For two load datasets $D$ and $D^{\prime}$ that differ only in one element, they hold the following adjacency relation $D\sim_{\mu}D^{\prime}$:

where $\mu$ is the scalar measuring the distance between two adjacent datasets.

For a non-private OPF algorithm, the corresponding outputs given two adjacent load datasets, $D$ and $D^{\prime}$, are distinguishable, leaving exploitable information for an adversary to reverse-engineer the original load dataset. Differential privacy defines the degree of privacy leakage in a probabilistic sense.

To integrate differential privacy into the learning process of VQC, we add Gaussian noise into gradients $\nabla_{\theta}\mathcal{L}(\boldsymbol{\theta},\boldsymbol{x}_{i})$ of parameterized quantum gates and clip the gradient of each sample by a pre-defined $\ell_{2}$ norm $C$.

Algorithm 1 outlines the differentially private VQC learning process. For each OPF sample, it encodes the input data into a quantum state, evolves it through the VQC, and measures the final quantum system to obtain the predicted OPF solution. After that, a gradient is calculated based on the prediction loss and then is clipped based on a norm $C$. By clipping, the influence of each individual sample on the gradient is bounded to protect privacy. Then, the Gaussian noises controlled by scale $\sigma$ are added to the clipped gradients before updating the VQC parameters $\boldsymbol{\theta}$. This process is repeated for multiple epochs, leading to a privacy-preserving VQC that can approximate OPF solutions while protecting individual load data.

The privacy guarantee of Algorithm 1 is described as the following theorem.

Proof: In each step of Algorithm 1, the noise following $\mathcal{N}(0,\sigma^{2}C^{2}\boldsymbol{I})$ are added on the average gradient of each batch. For each batch, we compute the per-sample gradients and clip them to differ at most $C$, which bounds the sensitivity of gradient $\Delta$ within $C$. By Gaussian Mechanism [13], if we choose $\sigma\geq\frac{\sqrt{2\log(1.25)/\delta}}{\epsilon}$, the algorithm is ($\varepsilon,\delta$)-differentially private. Since in each step, we sample one batch of size $B$ from $N$ data samples with the sampling rate $\frac{B}{N}$. From the theorem of privacy amplification by uniform subsampling [14], the resulting algorithm achieves ($\frac{B}{N}\varepsilon,\frac{B}{N}\delta$)-differentially privacy. Since there are $T$ epochs, we apply the privacy-preserving algorithm for $T$ times. From the advanced composition theorem [13], i.e., if the algorithm is composed of $T$ algorithms with each providing ($\varepsilon,\delta)$-differential privacy, the composed algorithm is $(\varepsilon^{\prime\prime},T\delta+\delta^{\prime\prime})$-differential private for any $\delta^{\prime\prime}>0$ with $\varepsilon^{\prime\prime}=\sqrt{2T\ln(1/\delta^{\prime\prime})}\cdot\varepsilon+T\varepsilon(e^{\varepsilon}-1)$.

Hence, the Algorithm 1, which composes $T$ steps with each providing ($\frac{B}{N}\varepsilon,\frac{B}{N}\delta$)-differentially privacy, achieves $(\varepsilon^{\prime},\frac{TB}{N}\delta+\delta^{\prime})$-differentially privacy. This completes the proof.$\hfill\square$

We first provide an illustrative example to demonstrate how an adversary detects the load pattern if the voltage measurements are exposed. Assume a customer at node 30 has an atypical load pattern representing the production technology. Its pattern under different timestep $t$ is formulated with the following three periodic components [7]:

The non-private OPF solution leaks the sensitive load information through the voltage readings, as displayed in Figure 3. The left figure presents the load pattern of the customer and the remaining figures show the voltage magnitudes at the same node. From left to right, in the second figure, we can see that the non-private OPF and non-private QNN demonstrate an informative “reverse” pattern of the load demand, disclosing the power consumption pattern of the customer. When the Gaussian noises are added to the training process, i.e., $\sigma\neq 0$, the solution from private QNN deviates from the non-private one and fluctuates more mildly to protect the load pattern. When noise with a larger scale $\sigma$ is introduced, the fluctuation is smaller, leading to stronger privacy protection.

To assess the accuracy of the probabilistic OPF solution generated by the private QNN, we computed the mean and standard deviation based on 1,000 samples. These samples were obtained by sampling the WT generation from a Weibull distribution $\textrm{Weibull}(1,4.8)$, PV generation from a Beta distribution $\textrm{Beta}(2,5)$, and load demand from a normal distribution $\mathcal{N}(0,0.3)$. These samples were then solved deterministically to collect the aggregate statistics.

Table 2 presents a comparison of the voltage magnitude at Node 30 across various perturbation levels ($\sigma=0$, $1$, $5$, and $10$). The reference values, i.e., the mean value of 12.6592 kV and standard deviation value of 0.0009 kV, were obtained via the Monte Carlo method. In the absence of noise ($\sigma=0$), the calculated statistics closely align with the reference values. As the noise scale increases, representing a greater level of privacy protection, the statistical results deviate more significantly from the reference mean. Specifically, the error percentage for the mean voltage remains within 0.0019%, indicating that the introduction of noise by the private QNN causes only minimal alterations to the voltage magnitudes, thereby allowing for accurate estimation of aggregate statistics. However, the standard deviation exhibits larger deviations as the noise restricts the model’s sensitivity to minor variations. Nonetheless, by selecting an appropriate noise scale, e.g. from $\sigma=0$ to $\sigma=1$, a balanced trade-off can be achieved between the accuracy in estimating the uncertainties and the privacy protection of the sensitive load patterns.

To demonstrate the advantage of quantum ML over classical ML, we implement a multilayer perceptron (MLP) and compare the accuracy between the proposed QNN and a classical MLP network. The MLP is developed using the PyTorch library and consists of two hidden layers, each containing 32 units. Both models are trained for 1,000 epochs. When noise is introduced, both models utilize Algorithm 1 to preserve the privacy of load demands. The evaluation metric is the coefficient of determination, i.e., $R^{2}$, which quantifies the degree to which a model’s predictions align with observed data, with values closer to 1 indicating a better fit.

Due to the limited availability of real quantum hardware, we adopt the formula proposed by Guerreschi et al. [16] to calculate the quantum computational time, including the execution time and the measurement time:

where $T_{p}$ denotes the time for state preparation, $T_{G}$ represents the average duration of a single quantum gate, $D$ is the maximal circuit depth (number of sequential quantum gates), and $T_{M}$ is the measurement time. The time scales are set as $T_{p}+T_{M}=1\ \mathrm{\mu}\mathrm{s}$ and $T_{G}=10\ \mathrm{ns}$, following the parameters in literature [17].

Table 3 presents a comparison of the accuracy in predicting the voltage magnitude at Node 30 across 1,000 samples. Under various noise scales, the QNN consistently outperforms the MLP. In the absence of noise, the obtained $R^{2}$ scores are comparable, with the QNN achieving 0.977 and the MLP achieving 0.971. However, when Gaussian noise is incorporated into the training process, the QNN demonstrates a clear advantage over the MLP. Specifically, at noise levels of $\sigma=1$ and $\sigma=2$, the MLP yields negative $R^{2}$ scores, whereas the QNN maintains positive $R^{2}$ values, indicating the MLP’s inability to effectively forecast the voltage magnitude under noisy conditions. This degradation is attributed to the accumulated noises, which hinder the MLP’s forecasting capability. Notably, at a noise scale of $\sigma=5$, the MLP exhibits a large negative $R^{2}$ value of -22,291.787, while the QNN still maintains a positive $R^{2}$ of 0.113. Furthermore, the QNN has shorter computation times and utilizes significantly fewer parameters, i.e., 165 versus 1,281, as shown in the second and third columns of Table 3, further highlighting the advantages of quantum ML over classical approaches.

To investigate the underlying reasons for the distinct performance between the QNN and MLP, we analyze the loss values during the training process. Figure 4 illustrates the case when $\sigma=1$. The graph reveals that the training loss of the MLP is highly unstable; although most loss values are around zero, some instances exhibit very large loss values. A potential reason is that the accumulated noises on gradients dominate the learning process and thus destroy the subsequent learning performance. In contrast, the QNN maintains stable training and is resilient to noise, highlighting the advantage of the QNN when incorporating differential privacy into the training process.